I'm going through a similar action myself right now -- I'm trying to kill as few humans as possible. (I made an exception for Benny, and raiders who attack on sight.)
Skills: I know you said you don't want to do locks and computers, but you'll have to for some things, so keep lockpicking and science high. Prioritize speech -- you can get out of a lot of fights, and even better, get XP for it. Maybe go for survival and/or repair, and go after the challenges for crafting.
So some early-game missions to help you out? Skip Primm entirely, head straight to Mojave Outpost. You'll get Ghost's quest there, which just requires you to go to Nipton. In Nipton, you'll get "Cold, Cold Heart," which just requires you to go to Mojave. Easy, non-violent 250 XP. Not much, but it'll get you started.
From there, make a beeline to New Vegas, by way of Novac. In Novac, do Boone's side quest, because it's quick and easy and non-violent. In Freeside, take the peaceful route through G.I. Blues, hook up the Followers and the Garretts, and do the Wang Dang Atomic Tango. Once in New Vegas, go talk to Mr. House and deal with Ring-a-Ding-Ding. After that, you should be able to take down Talent Pool pretty quickly and peacefully.
By the time you get through all this, with perhaps a little bit of minor side adventurism, and you should be at least level 10 or 12.
You can totally do this. I Mr. Sandman'd everyone, and killed off most of everyone in disguise so I barely lost any reputation. After I dealt with them all, I exited the bunker (without blowing it up and still completing the 'Destroy The Brotherhood' mission by killing everyone, or at least all major characters) and Veronica was none the wiser. All was good.
Best Answer
I found almost all of the timed explosives to be difficult to use and very situational. I only really used them, and only really tin grenades, in service of Crackerjack Timing as part of my 100% achievement run (which I documented here).
Most of the time enemies don't really have strong AI enough to plan where they're going to be in 15 seconds. (Well, besides the ones that will just run up in your face...) so setting an explosive and having exactly 15 seconds to bring someone into their radius is kind of hard to do. If someone is standing still or pathing in a consistent pattern, you might be able to measure out 15 seconds and catch them in the blast, but by that point you might as well just shoot them in the face (not the hands, not the legs, etc) and get it over with.
The one situation where it could be interesting is stealth reverse-pickpocketing a live bomb into an enemy's inventory. This will cause them to become a walking time bomb, and if they're pathing into an area with other hostiles, you might be able to take them all out at once.